EXAM 1
Problem 1.
is
$ x(t) = \sum_{k = -\infty}^\infty \frac{1}{(t+2k)^{2}+1} $
periodic?
We know that for a signal to be periodic
$ x(t) = x(t + T) $
So we shift the function by a arbitrary number to try to prove the statement above
$ x(t+1) = \sum_{k = -\infty}^\infty \frac{1}{(t+1+2k)^{2}+1} $
$ x(t+4) = \sum_{k = -\infty}^\infty \frac{1}{(t+2(\frac{1}{2}+k))^{2}+1} $
Then we set $ r = \frac{1}{2}+k $ to yield,
$ = \sum_{k = -\infty}^\infty \frac{1}{(t+2r)^{2}+1} $
Since this signal is equivalent to x(t), then x(t) is periodic.
